Reductions of Approximate Linear Programs for Network Revenue Management

The linear programming approach to approximate dynamic programming has received considerable attention in the recent network revenue management literature. A major challenge of the approach lies in solving the resulting approximate linear programs (ALPs), which often have a huge number of constraints and/or variables. We show that the ALPs can be dramatically reduced in size for both affine and separable piecewise linear approximations to network revenue management problems, under both independent and discrete choice models of demand. Our key result is the equivalence between each ALP and a corresponding reduced program, which is more compact in size and admits an intuitive probabilistic interpretation. For the affine approximation to network revenue management under an independent demand model, we recover an equivalence result known in the literature, but provide an alternative proof. Our other equivalence results are new. We test the numerical performance of solving the reduced programs directly using off-the-shelf commercial solvers on a set of test instances taken from the literature.